# Boundary conditions for $\mathbf D$ and $\mathbf H$

I understand the derivation for the boundary conditions for $$\mathbf B$$ and $$\mathbf E$$ as it was explained to me in Griffiths, but Griffiths states the following:

$$H_{\text{above}}^{\bot} - H_{\text{below}}^{\bot} = -(M_{\text{above}}^{\bot}-M_{\text{below}}^{\bot})$$

and

$$\mathbf D_{\text{above}}^{\parallel} - \mathbf D_{\text{below}}^{\parallel} = \mathbf P_{\text{above}}^{\parallel} - \mathbf P_{\text{below}}^{\parallel}$$

This feels like the right idea, due to the relations that $$\nabla \times \mathbf D = \nabla \times \mathbf P$$ and $$\nabla \cdot \mathbf H = - (\nabla \cdot \mathbf M)$$, and from what I learned of the implications by $$\nabla \cdot \mathbf E = \rho / \epsilon_0$$ implies a discontinuity in the parallel component of the electric field, which is why it makes sense to me that the discontinuity in the parallel component of the $$\mathbf D$$ field is the way it is. However, I can't quite piece together a coherent argument justifying this boundary conditions. How is this proven?

Also, the $$\mathbf H$$-field perpendicular component boundary conditions are expressed in magnitude in Griffiths, yet the $$\mathbf D$$-field parallel component boundary conditions were expressed as vectors, as I've shown. Why is that? If I made the $$\mathbf H$$-field a vector in the boundary condition I wrote for it, wouldn't that still be true?

• Born & Wolf, Principle of Optics, have derivation of the boundary conditions on the first few pages of the book – Cryo May 10 '19 at 21:39

In response to the second part of your question, I think this is because there is only a single perpendicular component to the H and M fields, call it z, while the parallel fields must be specified by two components, call it x and y, because the boundary spans a plane.

In brief, the argument goes:

$$\nabla \cdot {\bf D}$$ equation $$\rightarrow$$ condition on $$D_\perp$$.

$$\nabla \cdot {\bf B}$$ equation $$\rightarrow$$ condition on $$B_\perp$$.

$$\nabla \wedge {\bf E}$$ equation $$\rightarrow$$ condition on $$E_\parallel$$.

$$\nabla \wedge {\bf H}$$ equation $$\rightarrow$$ condition on $$H_\parallel$$.

The first two arguments involve a cylinder-shaped Gaussian surface. The second two arguments involve an integral around a loop hugging the boundary. A complete argument will include the time-derivative terms and show that they are negligible in the limit where the Gaussian surface or the loop closely hugs the boundary. A complete argument will also include the surface conduction current in the $$H_\parallel$$ result, and the surface density of free charge in the $$D_\perp$$ result.

Once you have the above then you can get the equations for things like $$E_\perp$$, $$D_\parallel$$, $$B_\parallel$$ and $$H_\perp$$ simply by using the definitions of $$\bf D$$ and $$\bf H$$ in terms of $${\bf E}$$, $${\bf P}$$, $${\bf B}$$, $${\bf M}$$.